By (REF ) we can write the conjugate cut locus \(Q(p)\)  as a disjoint union of the sets
\(A_1=\lbrace e_q(v)\in M: \: {\tau _{\text{cut}}}(p,v)={\tau _{\text{con}}}(p,v), \: \delta (v)=1 \rbrace \) 
and
\(A_2=\lbrace e_q(v)\in M: \: {\tau _{\text{cut}}}(p,v)={\tau _{\text{con}}}(p,v), \: \delta (v)\ge 2 \rbrace .\) 
To prove the claim of this proposition it suffices to show that
\(A_1\subset \lbrace e_q(v)\in N: \: \dim \left(\mathrm {D}e_q(T_{v}S_pM)\right)\le n-2 \rbrace ,\) 
since clearly we have that
\(A_2\subset \lbrace \exp _{p}(w)\in N: \: w\in T_pN, \: \dim \left(\mathrm {D}\exp _{p}(T_w(T_pN))\right)\le n-2\rbrace ,\) 
and therefore by the generalization of the classical Sard's theorem [1]} the Hausdorff dimension of \(Q(p)=A_1\cup A_2\)  is at most \(n-2\) .
We choose \(v_0 \in S_pN\)  such that \(e_q(v_0)\in A_1\) .
By the properties of the Jacobi fields normal to \(\gamma _{p,v_0}\) ,
we can identify the kernel \(\mathrm {D}\exp _{p}({\tau _{\text{con}}}(p,v_0)v_0)\)  with some vector sub-space of
\(T_{v_0}S_pM.\)
Since \(\dim T_{v_0}S_pM=n-1\)  we can verify the inclusion (REF ) if we show that
\(\ker \mathrm {D}\exp _{p}({\tau _{\text{con}}}(p,v_0)v_0)\subset \ker \mathrm {D}e_q(v_0).\) 
Since \(\delta (v_0)=1\)  we get by Lemmas REF  and REF  that there exists a neighborhood \(U\subset S_pN\)  of \(v_0\)  where the conjugate distance \({\tau _{\text{con}}}(p,\cdot )\)  and the map \(e_q(v)=\exp _{p}({\tau _{\text{con}}}(p,v)v)\)  are smooth. Let \(\xi \in T_{v_0}S_pM\)  be in the kernel of the differential of the exponential map. Then by the chain and Leibniz rules we get
\(\begin{split}\mathrm {D}e_q(v_0)\xi =\dot{\gamma }_{p,v_0}({\tau _{\text{con}}}(p,v_0))\mathrm {D}{\tau _{\text{con}}}(p,v_0)\xi .\end{split}\) 
Therefore \(\mathrm {D}e_q(v_0)\xi =0\)  if and only if \(\mathrm {D}{\tau _{\text{con}}}(p,v_0)\xi =0\) . So we suppose that \(\mathrm {D}e_q(v_0)\xi \ne 0\) .
By the Rank theorem [2]} we get that the subset of \(T_pN\) , near \({\tau _{\text{con}}}(p,v_0)v_0\) , where \(\mathrm {D}\exp _{p}\)  vanishes
is diffeomorphic to a smooth sub-bundle of \(TU\subset TS_pN\) . Then we use the existence of the ODE theorem to choose a smooth curve \(v(\cdot )\colon (-1,1) \rightarrow U\subset S_pN\)  such that \(v(0)=v_0\) , \(\dot{v}(0)=\xi \)  and \(\dot{v}(t)\in \ker \mathrm {D}\exp _{p}({\tau _{\text{con}}}(p,v(t))v(t))\)  for every \(t \in (-1,1)\) . Thus \(c(t):=e_{q}(v(t))\)  is a smooth curve in \(\hat{M}\)  that satisfies
\(\dot{c}(t)=\dot{\gamma }_{p,v(t)}({\tau _{\text{con}}}(p,v(t)))\frac{d}{d t}({\tau _{\text{con}}}(p,v(t))).\) 
Since \(\frac{d}{d t}({\tau _{\text{con}}}(p,v(t_0)))=\mathrm {D}{\tau _{\text{con}}}(p,v_0)\xi \)  we can assume that \(\frac{d}{d t}({\tau _{\text{con}}}(p,v(t)))>0\)  on some interval \((-\varepsilon ,\varepsilon )\)  for \(0<\varepsilon <1\) . Thus by equation (REF ) and the fundamental theorem of calculus we get that the length of \(c(t)\)  on \([-\varepsilon ,0]\)  is
\({\tau _{\text{con}}}(p,v_0)-{\tau _{\text{con}}}(p,v(-\varepsilon )).\) 
From here by the assumption \({\tau _{\text{con}}}(p,v_0)={\tau _{\text{cut}}}(p,v_0)\)  and the triangle inequality we get
\(\begin{split}{\tau _{\text{con}}}(p,v(-\varepsilon )) \ge &d_{\hat{M}}(p,e_q(v(-\varepsilon )))\ge d_{\hat{M}}(p,e_q(v_0))-\mathcal {L}(c)\ge {\tau _{\text{con}}}(p,v(-\varepsilon )),\end{split}\) 
and the inequality above must hold as an equality.
Therefore
\(d_{\hat{M}}(p,e_q(v(-\varepsilon )))+\mathcal {L}(c)= d_{\hat{M}}(p,e_q(v_0)),\) 
and the curve \(c(\cdot )\colon [-\varepsilon , 0]\rightarrow \hat{M}\)  is part of some distance minimizing geodesic \(\gamma \)  of \({\hat{M}}\)  from \(p\)  to \(e_q(v(0))\)  that contains \(e_q(v(-\varepsilon ))\) .
Thus we have after some reparametrization \(t=t(s)\)  that
\(\gamma (s)=e_q(v(t(s)))=c(t(s))\) 
for every \(t(s)\in (-\varepsilon ,0)\) . By (REF ) we get that \(\gamma \)  is a parallel to \(\gamma _{p,v(t)}\)  for every \(t \in (-\varepsilon ,0)\) . This is not possible unless the geodesics \(\gamma _{p,v(t)}\)  are all the same for every \(t \in (-\varepsilon ,0)\) . Hence \(v(t)\)  and \(c(t)\)  are constant curves. This leads into a contradiction.
The inclusion (REF ) is confirmed and the proof is complete.
